The growing library of two-dimensional layered materials is providing researchers with a wealth of opportunity to explore and tune physical phenomena at the nanoscale. Here, we review the experimental and theoretical state-of-art concerning the electron spin dynamics in graphene, silicene, phosphorene, transition metal dichalcogenides, covalent heterostructures of organic molecules and topological materials. The spin transport, chemical and defect induced magnetic moments, and the effect of spin-orbit coupling and spin relaxation, are also discussed in relation to the field of spintronics.